According to Theorem 4.7 in [1]}, we have that
\(\widetilde{\overline{U}}_{h}\rightarrow u \qquad \text{in $C([0,T];L_\textup {loc}^r(\mathbb {R}^N))$ as $h\rightarrow 0^+$,}\) 
and \(u\in (L^1\cap L^\infty )(Q_T)\cap C([0,T];L_\textup {loc}^r(\mathbb {R}^N))\) . Note that, for all \(t,s\in [0,T]\) , we use Proposition REF  to obtain
\(\begin{split}&\big \Vert \big (u(\cdot ,t)-u(\cdot ,s)\big )\mathbf {1}_{|\cdot |\le R}\big \Vert _{L^r(\mathbb {R}^N)}\\&=\big \Vert \big (u(\cdot ,t)-\widetilde{\overline{U}}_{h}(\cdot ,t)\big )\mathbf {1}_{|\cdot |\le R}\big \Vert _{L^r(\mathbb {R}^N)}+\big \Vert \big (\widetilde{\overline{U}}_{h}(\cdot ,t)-\widetilde{\overline{U}}_{h}(\cdot ,s)\big )\mathbf {1}_{|\cdot |\le R}\big \Vert _{L^r(\mathbb {R}^N)}\\&\quad +\big \Vert \big (\widetilde{\overline{U}}_{h}(\cdot ,s)-u(\cdot ,s)\big )\mathbf {1}_{|x|\le R}\big \Vert _{L^r(\mathbb {R}^N)}\\&\le 2\big \Vert \big (u-\widetilde{\overline{U}}_{h}\big )\mathbf {1}_{|\cdot |\le R}\big \Vert _{C([0,T];L^r(\mathbb {R}^N))}+\sup _{0<h<1}\big \Vert \widetilde{\overline{U}}_{h}(\cdot ,t)-\widetilde{\overline{U}}_{h}(\cdot ,s)\big \Vert _{L^r(\mathbb {R}^N)}\\&\le 2\big \Vert \big (u-\widetilde{\overline{U}}_{h}\big )\mathbf {1}_{|\cdot |\le R}\big \Vert _{C([0,T];L^r(\mathbb {R}^N))}+\Lambda (|t-s|)+\int _{s}^{t}\Vert g(\cdot ,\tau )\Vert _{L^r(\mathbb {R}^N)}\,\mathrm {d}\tau .\end{split}\) 
Now, send \(h\rightarrow 0^+\) , and then, by the Fatou lemma, \(R\rightarrow \infty \) . This gives
\(\Vert u(\cdot ,t)-u(\cdot ,s)\big \Vert _{L^r(\mathbb {R}^N)}\le \Lambda (|t-s|)+\int _{s}^{t}\Vert g(\cdot ,\tau )\Vert _{L^r(\mathbb {R}^N)}\,\mathrm {d}\tau \rightarrow 0\qquad \text{as $|t-s|\rightarrow 0$,}\) 
and the desired regularity is obtained.
